According to [1]}, the space of stability conditions \(\operatorname{Stab}(\mathcal {C})\) 
is a complex manifold of dimension \(\operatorname{rk}\Gamma \) , such that the map \(\operatorname{Stab}(\mathcal {C})\rightarrow \operatorname{Hom}(\Gamma ,{C})\)  which sends \(\sigma =(Z,\mathcal {A})\mapsto Z\)  is a local homeomorphism of
complex manifolds. Moreover, it admits an action of \(\widetilde{GL^+}(2,{R})\times \operatorname{Aut}(\mathcal {C})\) 
[1]},
where \(\widetilde{GL^+}(2,{R})\)  is the universal cover of the group of \(2\times 2\)  real matrices with
positive determinant. The group \(GL(2,{R})^+\)  acts on the central charge \(Z\)  via
\(\begin{pmatrix} \Re Z \\ \Im Z \end{pmatrix} \mapsto \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} \Re Z \\ \Im Z \end{pmatrix}\ ,\quad ad-bc>0\) 
